TOPIC 1: ELECTROMAGNETISM | PHYSICS FORM 6

TOPIC 1: ELECTROMAGNETISM | PHYSICS FORM 6

This is the production of a magnetic field by current flowing in a conductor.

The magnetic effect of current was discovered by Ousted in 1820. The verified magnetic effect of current by the following simple experiment.​​

Figure below shows a conducting wire AB Above a magnetic needle parallel to it.

So long as there is no current in the wire, the magnetic needle remains parallel to the wire i.e. there is no deflection in the magnetic needle.​​

As soon as the current flows through the wire AB, the needle is deflected.


​​ Magnetic needle

When the  current in wire  AB  is  Reversed the  needle is  deflected  in the  opposite  direction​​

This  Deflection is  a convincing proof of  the  existence  of  a magnetic field  around  a  current  carrying conductor.

On increasing the current in the wire AB the deflection of the needle is increased and vice versa.

This  shows  that  magnetic field  strength  increases  with  the  increase in  current and  vice versa  

It is clear from Worsted’s experiment that current carrying conductor produces a magnetic field around it.​​ 

The  larger the  value  of  current in the  conductor the  stronger is the  magnetic  field and  vice  versa.

Magnetic field

Is the  region around a magnet where magnet effect can be experienced.​​

OR

Is the space around a current carrying conducting (magnet) where magnetic effects can be experienced.​​

The  Direction of  a field at a point is  taken to be  the  direction in  which  a  north magnetic pole would  move more  under  the  influence  of  field  if it  were placed at  that point.

The magnetic field is represented by magnetic lines of force which form closed loops.​​

The magnetic field disappears as soon as the current is switched off or charges stop morning.

Magnetic  flux is  a  measure  of  the number  of  magnetic field  lines passing  through the  region.

The  unit of  magnetic flux is  the  Weber (Wb)
The  flux through an  area  A  on figure  below  the   normal  to  which  lies  at  angle 𝜃 ​​ to ​​ a ​​ field ​​ of ​​ flux ​​ density B​​ 
 ​​  ​​  ​​  ​​           ​​ 
Is  a  quantity  which  measures the  strength  of  the  magnetic field​​

It  is  sometimes  called magnetic  ​​

It is  a vector  quantity​​

The  SI unit  of  Magnetic  flux density is  Tesla (T) or  Wb/m2

Magnetic flux density is simply called magnetic field B​​

B = θ/A

FORCE ON A MOVING CHARGE IN A MAGNETIC FIELD

Consider a positive charge +Q moving in a uniform magnetic field ​​ with a velocity​​ 
Let the  Angle between ​​ and ​​ be θ  as  shown​​ 


​​ It  has  been found experimentally  the  magnetic  field  exerts a  force  F on  the  charge.

The  magnitude  F of  this  force  depends  on the  following factors​​ 
(i) F  α θ
(ii)F  α B
(iii)​​ Combining the factors we get​​ 
​​ 

Where K is a constant of proportionality​​

The unit of B is so defined that K = 1​​ 
Equation (a) can be written in a vector form as:-
​​ F = the force of the particle (N)​​ 
B = the magnitude of the magnetic flue density of the field T
Q = the charge on the particle
V= the magnitude of the velocity of the particle​​ 

Definition of ​​ From
F = BQVsin⁡θ​​  
  ​​  If V = 1, Q = 1,​​ θ= 90 then

F = Sin90
F=B     ​​ 

Magnetic field (​​ ) at a point in space is equal to the force experienced by a unit charge moving with a unit velocity perpendicular to the direction of magnetic field at that point  

Right Hand Grip Rul

Grip the wire  using  the  right hand with the  thumb  pointing in the  direction of the  current  the  other fingers unit point in the  direction of  the  field.

–          For an electron (negatively charged) entering the magnetic field as shown below

 

The Direction of positive charge will be exactly opposite. Applying Right hand Grip Rule it is clear that  Direction  of force on the  electron will be  vertically  upward

For a  positively charged particle, it will be  vertically downward

Direction of magnetic field means from N –pole to S-pole.

SOME CASES OF MAGNETIC FORCE F

Consider an electric charge Q moving with a velocity V through a magnetic field B. then the magnetic force F on the charge is given by

                               F = BQV

(i)        When ​​  = 0o​​ or  1800  

F = BQV

F=   BQVSin00​​ or   F = BQV

 F= 0                       ​​ 

Hence  a charged particle moving parallel(or Anti parallel) to the  direction of magnetic field experiences no force

(ii)       When ​​  =900

 F = BQV

F = BQV​=1

 F = BQV

Hence a force experienced by charged particle is maximum when it is moving perpendicular to the direction of magnetic field.

(iii)    When V=O, the charge particle is at rest.

F = BQV

F=BQ(0)

∴F=O​​ 

If a charged particle is at rest in a magnetic field it experiences no force.

(iv)   When Q = O​​ 

 F = BQVF = 0​​ 

Hence electrically neutral particle (eg neutron) moving in a magnetic field experiences no force. 

The magnetic force F acts perpendicular to velocity V (as well as B)

This  means  that  a uniform magnetic field  can  neither  speed  up  nor  slow down a  moving charged particle;  it  can  charge only the  Direction  of V and  not  magnitude of  V​​ 

Since  the  magnitude  of  V does  not  charge the  magnetic force  does not change  the  kinetic energy of  the  charged particle.

UNITS AND DIMENSIONS OF​​

 The SI unit of magnetic field B is​​ Tesla

            Now​​ 

            F = BQV

            ​​ 

If   Q = 1C, V =1m/s, Q= 900   F= 1N​​ 

B  = 1T​​ 

 Hence   the  strength of  magnetic field  at a  point is  1T if  a charge of  1C when  moving  with a  velocity of  1m/s  at  right angles to  the  magnetic field, experiences a  force  of  1N at  that  points.

Magnetic field of earth at surface is about 10​​ – 4T. On the other hand, strong electromagnets can produce magnetic fields of the order of 2T.

Dimensions of​​ 

Worked Examples

1.       A proton is moving northwards with a velocity of​​ m/s in a magnetic field of 0.1Tdirected eastwards. Find the force on the proton. Charge on  proton = 1.6 x 10​​ -19C.

Solution

 F = BQV

B= 0.1T

 V=​​ m/s
F=0.1 X 1.6 X 10-19​​ X 5 X 106X Sin 90​​ 

Q = 1.6 X 10-19C

 = 900

2.  An electron experiences the greatest force as it travel   at 3.9 x105​​ m/s in a magnetic field when it is moving westward.  The force   is upward and is of magnitude​​ N what is the magnitude and direction of the magnetic field.

Solution

The conditions of the problem suggest that the electron is moving at right angles​​ 

                 ​​  To the direction of the magnetic field

               F = BQV, F = 8.7 x 10​​ -13N

                Q= 1.6 X10​​ -19C

                 V=3.9X105m/s

              B = 13.14T

By right hand rule per cross product, the direction of the magnetic field is towards northward.

3.    An  α  – particle of mass 6.65 x 10-27​​ kg is  travelling at right angles to a magnetic field with a speed of 6×105m/s. The strength of   the magnetic field is 0.2T.calculate the force on the​​  – particle and its acceleration.

 Solution
  Force on   α – particle   F = BQV

  M = 6.65 X10-27Kg​​ 

  V = 6 x 105m/s

   B = 0.2T

  ​​  = 900​​ 

    F = BQV

    = (0.2 x 2x 1.6×10-19) x​​ x Sin90Ëš
​​ 
​​ 

      Acceleration of α – particle​​ 

     F= mÉ‘​​ 

   É‘=​​ =   

4.       A  copper  wire  has  1.0 x 1029​​ free  electrons per  cubic meter, a  cross sectional  area  of  2mm2​​ and  carries  a  current of  5A  . The wire is placed at right angle to a uniform magnetic field of strength 0.15T. Calculate the force the acting on each electron.

Solution

        ​​ I = neA

Drift velocity =​​ n= 1×1029m-3   e = 1.6×10-19c   A= 2mm2​​ = 2×10-6m2

         I = 5A​​ 

Force on each electron F= BQ Sin

Q= 1.6 x 10-19c

B= 0.15T                                             ​​ 

Q=900

BIOT – SAVART LAW

The  Biot – Savart  law  states  that the  magnitude  of  magnetic  flux  density  dB  at a point  P  which is  at a distance  r  from  a very short  length  dl of  a conductor  carrying  a current I  is  given by.​​ 

where​​  is  the  Angle between the  short length dl and  the  line  joining  it to point  P

          K  is a constant of  proportionality  its  value  depends on the  medium in which the  conductor is  situated and  the  system of units  adopted.

           For  free space  vacuum  or air​​ 

 

This equation is known as Biot –Savart Law and gives the magnitude of the magnetic field at a point due to small current element

Current element

 Is  the  product of  current (I)  and  length of  very small  segment () of  the  current carrying  conductor.

Current element =

Current element produces magnetic field just as a stationary charge produces an electric field the current element is a vector.

Its  Direction is  Tangent  to the  element and  acts in the direction of  current flow  in the  conductor​​ 

Biot -Savart law holds strictly per steady currents

Direction of     B

The  direction  of    is  perpendicular  to  the  plane  containing      and        by  right hand rule  for the  cross  product the  field  is  directed inward. ​​ 

Special cases​​ 

           ​​ 

(i)   When​​  = 00​​ or 1800

i.e Point P lies on the axis of the conductor

  ​​ 

Hence there is no magnetic field at any point on the thin current carrying conductor minimum value.

(ii)    When​​  = 900

When point P lies at a perpendicular position​​ w .r. t​​ current element

Hence magnetic field due to a current element is maximum in a plane passing through the element and perpendicular to its axis.

Important point about Biot – Savant law

(i)   Biot – Savant law is valid per symmetrical current distributions.
​​ 
(ii)    Biot – Savant  law  cannot  be  proved  experimentally because  it is  not  possible to have  a current  carrying  conductor  of  length dl

(iii)     Like  coulomb’s  law  in  electrostatics, Biot- Savant law  also obeys  inverse square  law

(iv)     The  Direction of  dB   is  perpendicular to  the  plane  containing    and ​​ 

(v)      This  law  is  also  called  Laplace’s  law and  inverse square law’

 BIOT – SAVART LAW VERSUS COULOMB’S LAW IN ELECTROSTATICS

 According  to  coulomb’s  law  in  electrostatics, the  eclectic field due to  a  charge  element dQ  at a distance  r is  given by​​ 

       ​​ 

According to Biot – Savart law the magnetic field due to a current element    at a distance r is given by  ​​ 

From the above two equations we note the following points of Similarities and Dissimilarities.

Similarities​​ 

(i)   Both laws obey inverse square  law​​ 

(ii)  Both the  fields(magnetic field and  Electro static field) obey  superposition principles

(iii)Both the fields are long range fields.

Dissimilarities​​ 

(i)  The Electric field is produced by a scalar source i.e.  Electric charge. However the magnetic field is product by a vector source i.e.  current  

(ii)  The Direction the Electric field is along the displacement vector i.e.  The line joining the source and field point. However  the  direction of  magnetic  field  is  perpendicular  to the  plane  containing current  element ​​ and  displacement vector  

(iii)    In Biot –Savant law the magnitude of magnetic field dB α Sin Where​​  is the  Angle  between current element   and  displacement vector     However there is  no  angle  dependence in  coulomb’s law for electrostatics

MAGNETIC FIELD AT THE CENTER OF CURRENT CARRYING CIRCULAR COIL

 Consider a circular coil of radius r and carrying current I in the Direction shown in figure​​ 

 

                                                                  ​​ 

Suppose  the   loop  lies  in the  plane  of paper it is  desired  to find  the  magnetic field  at  the  centre O of  the  coil​​ 

Suppose  the  entire  circular coil is divided into a  large  number  of  current  elements each of  length​​

According  to  Biot – Savant  law, the  magnetic field ​​ at the centre O of the  coil  due  to current  element​​  is  given  by​​ 

……………

The  direction  of  dB  is  perpendicular to the  plane  of the  coil and is  Directed  inwards

Since  each  current  element  contributes  to the magnetic field  in the  same  direction, the  total magnetic field B  at the  centre O can be  found  by integrating equation…………(i)​​ 

              L- Total length of the coil = 2r​​

 

 If the coil has N turns each carrying current in the same direction then contribution of all turn are added up.

 B=

MAGNETIC FIELD DUE TO INFINITELY LONG CONDUCTOR

 

 The flux density dB at P due to the start length dl given by equation as

From the figure (A)​​ 

      ​​    ,​​ 
​​       ​​ 

      r =  ​​                         ​​ 

       ​​  = a cot​​ 

      ​​  = -a 

Substituting for​​  and​​  gives

       ​​ 

  The total flux density B at P is the sum of the flux densities of all the short lengths and can be found by letting d→O and integrating over the whole length of the conductor.

      ​​ 

 The  limits  of  the  integration  are​​  and 0 because  these are  values of 𝜃​​ at the ​​ ends of the ​​ conductor
    ​​   ​​ 

       ​​ 

       ​​ 

      ​​ 

FLUX DENSITY AT ANY POINT ON THE AXIS OF A PLANE CIRCULAR

Circular coil with its plane perpendicular to that of the paper

The  flux  density dB at p due  to the  short length dl of the coil  at  X, where  X is  in  the  plane  of the  paper, is  given by  equation as​​ 

   ​​ 

          By symmetry, when all the short lengths​​  are taken into account the components of magnitude​​  sum to zero.

          Each  short length  produces a component of magnitude​​ Sin α parallel to the  axis and  all those components are  in the  direction shown

          The  total  flux density  is  therefore  in  the  direction of​​ Sin α  and  its magnitude B is  given by​​ 

             ​​ 

          ​​ 

 The radius vector XP of each small length is perpendicular to it, so that​​ =900​​ and there pore Sin = 1

        ​​ 

Since,

= 2(the circumference of the coil)

 

 , But​​  =​​ 

 

 

For a coil of N Turns

 

 When S= r​​ 

Also from the figure

 AMPERE’S CIRCUITAL LAW

States that the line integral of magnetic field    around any closed path in vacuum/air is equal to​​ times the total current (I) enclosed by that path​​ 
     ​​ 
I = current enclosed by that path.                                 

Ampere’s  law is  an  alternative  to  Biot –  Savart law  but  it is  useful for  calculating  magnetic field  only in situations with considerable symmetry.

This law is true for steady currents only.

In order  to  use  law  it is  necessary  to  choose  a  path  for which it  is possible  to determine the  value of  the  line  integral

It  is  because  there  are  many  situations where there  is  no such path  that  the law is of  limited use.

 Hence the application of ampere law​​

(i) Magnetic field due to constraining conductor carrying current
(ii)Magnetic field due to solenoid carrying current
(iii)Magnetic field due toroid

MAGNETIC FIELD DUE TO STRAIGHT   CONDUCTOR CARRYING CURRENT

 Consider a long straight conductor carrying current I in the direction as shown in the figure below

It is desired to find the magnetic field at a point p at a perpendicular distance r for the conductors

Applying Ampere’s circuital law to this closed path​​ 
SOLENOID​​ 

Is a long coil of wire consisting of closely packed loops

Or

 Is a cylindrical coil having many numbers of turns ​​ 

By  long  solenoid we  mean that  the  length of  the Solenoid is very large as  compared to  its  Diameter.

Figure  below  shows the  magnetic field lines due to an  air cored solenoid carrying current

Inside the solenoid the magnetic field is uniform and parallel to the solenoid axis.​​ 

Outside  solenoid  the  magnetic field is  very  small as  compared  to the  field inside  and  may be  assumed  zero.

It  is  because the  same  no  of  field  line  that  are  concentrated  inside the  solenoid spread out  into very  faster  space  outside​​ 

 Magnetic flux density due to an Axis of an in finely long Solenoid

Consider  the  magnetic  flux  density​​  at P  due  to  a section of the  solenoid of  length ​​ 

 n = number of turns per unit length.

N= number  of  turns  the  section can be  treated  as a plane  circular coil of  N turns  in  which  case  dB is  given by​​ 

 

Since​​ dx​​ is small, the section can be treated as a plane circular coil or N turns in which case dB is given by           ​​ 

From the figure​​ 

 Also

Substituting for​​  and dx gives,

The  flux  densities   at  P due to  every  section  of the  Solenoid  are  all  in the  same  direction  and  therefore  the  total  flux  density  B can  be  found by  letting  dB→o and  integrate over  the  whole  length of the  solenoid.​​ 

The limits of integration are​​  and 0 because these values of β at the end of the solenoid.​​ 

 

 

If the Solenoid is Iron-cored of relatively permeability​​  magnitude of magnetic field inside the Solenoid is​​ 

From

At points near the ends of an air cored Solenoid, the magnitude of magnetic field is​​ 

 

 

 

The magnetic field outside a solenoid is zero

Also in a current carrying long solenoid the magnetic field produced does not depend upon radius of the Solenoid.

TOROID

Toroid is a solenoid that bent into the form of the closed ring.

The magnitude field B has a constant magnitude every where inside the toroid while it is zero in the open space interior and exterior to the toroid.

If any closed path is inside the inner edge of the toroid then ther is no current enclosed. Therefore, by Ampere’s circular law B=0.

Magnetic field ​​ due to toroid
Consider the diagram below
                    ​​ 
Let r = mean radius of toroid
I = Current through toroid
n = number of turns permit length
B = magnitude of magnetic field inside the toroid


Then

 

TOPIC 1: ELECTROMAGNETISM II | PHYSICS FORM 6

FORCE ON A CURRENT CARRYING CONDUCTOR PLACED IN A MAGNETIC FIELD

We know that a moving charge in a magnetic field experiences a force

Now electric current in a conductor is due to the drifting of the force electrons in a definite direction in the conductor

When such a current carrying conductor is placed in a uniform magnetic field, each free electron experiences a force.

 Since the free electrons are constrained in the conductor, the conductor itself experiences a force.

          Hence a current carrying conductor placed in magnetic field experiences a force F.

 Consider a conductor of the length L and area of cross- section a placed at an angle 𝜃​​ to the direction of uniform of magnetic field B.

– is the angle between the plane of the conductor. The magnetic force experienced by the moving charge in a conductor is F = BQV Sin

            For steady current       I =​​ 

                                                Q =I t

                                                F=​​ 

The velocity for direct current is constant

                                                V=​​ 

                                                F = B I t

                                               ​​ 

F= Force on the conductor (N)

B= Magnitude of the magnetic flux density of the field (T)

I = Current in the Conductor (A)​​ 

L= length of the conductor (M)

  The current in the conductor I​​ 

Special cases​​ 

  

Thus if current carrying conductor is placed parallel to the direction of the magnetic field of the conductor will experience no force.

ii.)​​ 

                        F = BIL

                     ​​ 

Hence current carrying conductor will experience maximum force when it placed at right angles to the direction of the field.

One Tesla​​ 

Is the  magnetic  flux density  of a  field  in  which a  force of  IN  acts on  a 1M length of a  conductor  which is  carrying a  current of  IA and  is  perpendicular to  the  field.​​ 

 B = Tesla​​ 

 The Direction of the force​​ 

Experiment shows that  the  force  is  always perpendicular  to the  plane  which  contains both the  current and  the  external field   at the  site of  the  conductor​​ 

The  direction  of the  force  can be  found by  using  Fleming’s left  hand rule​​ 

 Fleming’s left hand rule​​ 

States that if the first and the second fingers and the thumb of the left hand are placed comfortably at right angles to each other, with the first finger pointing in the direction of the current then thumb points in the direction of the force i.e. Direction in which Motion takes place If the conductor is free to move.

Maxwell’s Corkscrew rule

States that if a right handed corkscrew is turned so that its point travels along the direction, the direction of rotation of corkscrew gives the direction of the magnetic field.

FORCE BETWEEN TWO PARALLEL CONDUCTORS CARRYING CURRENTS

When two parallel current carrying conductors are close together, they exert force on each other.

It is because one current carrying conductor is placed in the magnetic field of the other

If currents are in the same direction the conductor attract each other and  If currents are in the opposite directions conductors  repel each other

Thus like currents attract, unlike currents repel.

Consider two infinitely long straight parallel conductors X and Y carrying currents I1​​ and I2​​ respectively in the same direction.

 Suppose the conductors are separated by a distance​​ rin the plane of the paper.

As each conductor is in the magnetic field produced by the other, therefore each conductor experiences a force

The current carrying conductor Y is placed in the magnetic field produced by conductor X

Therefore force act on the conductor Y.  The magnitude of the magnetic field at any point P on the conductor Y due to current I, in the conductor X is

By  right  hand  grip rule ; the  direction of  B  is  perpendicular  to the  place  of the  paper  and  is  directed  inwards.

 Now conductor Y carrying current​​  is placed in the magnetic field​​ produced by conductor X

         Therefore force per unit length of conductor  Y will experience a force​​  given by​​ 

 =​​ 

 According to FLHR, force​​  on conductor Y acts in the place of the paper perpendicular to Y and is directed towards to the conductor X.

Similarly, the Force on conductor X per unit length is  ​​ = ByI1L

 

But

Hence when two long parallel conductors carry currents in the same direction they attract each other. The force of attraction per unit length is      ​​ 

 

 

This shows that the attraction between two parallel straight conductors carrying currents in the same direction in terms of magnetic field lines of conductors

It is clear that in the space between X and Y the two fields are in opposition and hence they tend to cancel each other​​ 

However in the space outside X and Y the two fields assist each other. Hence resultant  field  distribution will be​​ 

If  two straight  current  carrying  conductors of  unequal length  are  held parallel to each other  then force  on the  long  conductor  is  due to the   magnetic field  of the  short conductor​​ 

I1​​ = Current through short conductor​​ 

l​​ = Length of short conductor​​ 

I2​​ = Current through long conductor​​ 

L = Length of long conductor​​ 

If r is the separation distance between these parallel conductors​​ 

Force on Long conductor = force on short conductor​​ 

 
Force on each conductor is the same in magnitude but opposite in direction (Newton’s third law)

DEFINITION OF AMPERE ​​ 

Force between two current currying conductors per unit length​​ 

If ​​  And r =1m then

 Ampere​​ 

 Is  that  steady  current  which when it is  flowing  in  each  of  two infinitely  long, straight parallel  conductors  which  have  negligible  areas  of  cross – section  and  are  1m apart  in a vacuum, causes each conducts to  exert  a force  of ​​ N on each mete  of the  other. ​​ 

 WORKED EXAMPLES

1. The  plane  of a  circular  coil  is  horizontal  it  has  20 turns  each of  8cm radius A current  of  1A flows  through it  which  appears  to be  clockwise from a point vertically  above  it. Find the  Magnitude  of the  magnetic  field  at the  centre of the  coil.

Solution

The  magnitude  of the  magnetic  field  at the  centre of the  coil  carrying  current  is  given by,
As the currents appears to be clockwise from appoint vertically above the coil the direction of the field will be vertically downward (By R.H.G.R)

 ​2. A wire placed along the South-North direction carries currents of 5A from South to North. Find the magnetic field  due to a 1cm piece of wire at a point 200cm North-East from the place.

Solution

By RHGR, The field is vertically vertical downwards​​ 

3. A coil of radius 10cm and having 20 turns carries a current of 12A in a clockwise direction when seen from east. The coil is in North – South plane.  Find the magnetic field at the centre of the coil.                                                 ​​ 

Solution

The magnitude of the magnetic field at the centre of the coil​​ 

The electron of hydrogen atom moves along a circular path of radius 0.5 x 10-10​​ with the uniform speed of 4 x 106​​ m/s.  Calculate the magnetic field produced by electron at the centre ( e= 1.6 x 10-9c)​​ 

Number the revolution made by the electron in 1 second is

                                      ​​ 

Current   =​​ 

I =  ​​ 1.27 X 1016​​ X 1.6 X10-19

                        1S

           I =     2.04 X 10-3A

 Magnetic field produced by the electron at the centre is​​ 

5.  A circular  coil  of  100 turns  has  a radius of  10cm and  carries  a current of  5A Determine  the  magnetic  field​​ 

(i)  At  the  centre  of  the  coil​​ 

(ii)  At a point  on the  axis  of  the  coil  at  a distance  of  5cm from the  centre  of the  coil.​​ 

Solution

(i)   Magnetic field  at the  centre  of the  coil is​​ 

 =   4 x 10-7​​ TA​​ -1

N    = 100 turns

 I = 5A

 r = 10×10-2m​​ 

B =​​ 4 x 10-7x 100 x S

2 X 0.1​​ 

B=   3.14 X10-3​​ T                The magnetic field of the centre of the coil  =   3.14 X10-3​​ T

(ii)    Magnetic  field  on the  axis  of the  coil  at  a  distance  X from the  centre is ​​ 

  = 4 x 10-7​​ TA​​ -1​​ 

 N   = 100 turns

  I    = 5A

  r    = 10 x 10-2

  x    = 0.05m​​ 

6. An electric  current  I   is  flowing  in  a  circular  wire  of  radius  at  what  dose  from the  centre  on the  axis  of  circular wire  will the  magnetic field  be  1/8th​​ of its  value  at  the  centre?

Solution

Magnetic field B at the centre of the circular coil is​​ 

Suppose at a distance X from the centre on the axis of the circular coil the magnetic field is ​​ 

 

 

 

 

 

 

 

        7. In  Bohr’s  model of  hydrogen  atom  the  electron  circulates  around  nucleus on a  path of radius  0.51Å at  a  frequency  of  6.8x is  rev/second  calculate  the  magnetic field induction at the  centre of the  orbit.

Solution

 The circulating electron is equivalent to circular current loop carrying current I given by​​ 

 

    ​​ 

    ​​ 

    I = 1.6​​ 

     I = 1.1A

 

Magnetic field at the centre due to this current is​​ 

 

 

 

                           = 14T​​ 

 

      8. A long straight wire carries a current of 50A. An  electron  moving at  107ms is  5cm from  the  wire​​ 

                                 ​​ 

 Find the Magnetic field acting on the electron velocity is directed​​ 

(i)    Towards the wire

(ii) Parallel to the  wire​​ 

(iii)  Perpendicular  to the  directions  defined  by  I and  ii

 

 Solution ​​ 

The magnetic field produced by current carrying long wire at a distance r

 

 

 The field is directed downward perpendicular   to the plane of the paper​​ 

( i)    The velocity V1​​  is towards the wire. The  angle between  VI  and  B is  900​​ force on  electron​​ 

F= BQV

F = 2x 10-4​​ x 1.6×10-19x107x Sin 900​​ 
F = 3.2 x 10-16​​ N

 

(ii)  When  the  electron  is  moving  is  moving  parallel to the  wire ,angle  between  V2 and B is  again  90Ëš Therefore, force  is  again ​​ 
3.2×10-16N

 

(iii)   When  the  electron is  moving perpendicular to the  directions  defined by  (i) and (ii) the  angle  between  V and B is O

F = O

   ​​ 

       9. A solenoid  has  a length  of  1 .23 m and  inner diameter  4cm it  has  five  layers of  windings  of  850 turns each and  carries  a current of  5.57A. what  is  the  magnitude  of the  magnetic  field  at the  centre  of the  solenoid​​ 

 

 

Solution

The  magnitude  of the  magnetic  field at the  centre  of  a  solenoid  is  given  by
​​ 

But

    10.   A  to void has a  core ( non –  ferromagnetic) of  inner  radius  20cm and  over  radius  25cm around which 1500 turns  of a wire  are  wound. If  current  in the  wire  is  2A

Calculate the magnetic field​​ 

(i)                  Inside  the  to void ​​ 

(ii)               Outside the  to void​​ 

 

Solution

( i)   The  magnitude  of the  magnetic  field  inside  the  toroid is  given  by​​ 

2
  ​​ 
    B = 0.003T

(ii)The magnetic field outside the toroid is Zero. It is all inside the toroid.

    11.   A solenoid 1.5m long and 4cm in diameter possess 10 turnscm. A current of 5A is flowing through it. Calculate the  magnetic  induction

(i)   Inside  and​​ 

(ii)  At  one end  on the  axis  of the  solenoid

 ​​ 

Solution
n = =​​  =​​ 

 

(i)   ​​ Inside the solenoid , the magnetic induction is given by

B =​​ 

                B = 4

B =​​ 

 

(ii)    At the  end  of the  solenoid  the  magnetic  induction is  given  by​​ 

 

     12.    (a)  How will the magnetic field intensity at the centre of a circular loop carrying current change, if the current through the coil is doubled and the radius of the coil is halved?

 

 (b) A long wire first bent in to a circular coil of one turn and then into a circular​​ 

          coil    of  smaller radius having  n  turns, if the  same  current passes in both the  cases, find  the  ratio of  magnetic  fields produced at the  centers in the  two cases.

 

(c) A and B are  two concentric coils of centre O and carry currents IA​​ and IB  as shown in figure​​ 

                               ​​ 

If the ratio of their radii is 1:2 and ratio of flux densities at O due to A and B is 1:3, find the value of​​ 

         Solution

(a)      Magnetic field at the centre of circular coil

 

 

 

 

 

 

         ( b)Suppose r is the radius of one turn coil and the r1​​ is the radius of n-turn coil. Then​​ 

N

 

 

First case                                  Second case                                        ​​ 

                                  ​​ 

 

 

 ​​ 

Solution
C.  Magnetic field at the centre of circular coil
​​ 

 

     13.   A helium nucleus makes a full rotation in a circle of radius 0.8m in two seconds. Find the value of magnetic field at the centre of the circle.​​ 

 

Solution​​ 

The charge on helium nucleus

Q=​​ e

Q=​​   1.6 X10-19c

Current produced I =​​ 

I =​​ 2 x 1.6 x 10-19

2

I =1.6 x10-19A

Magnetic field at the centre of the circle orbit of the helium is,

  ​​ 
B=​​ 

       ​​ 
​​ B= 1.256 x 10-25T  ​​ 

     14.   A soft Iron ring has a mean diameter of 0.20m and an area of cross section 5×10-4m2​​ it is uniformly wound with 2000turns carrying a current of 2A and the magnetic  flux in the iron is 8x 10-3Wb. What is the relative permeability of iron?​​ 

 

Solution​​ 

Length of ring l

l = 2

l = 2 x 0.10m

 

Number of turns per unit length n​​ 

n =​​ =​​ 

 

If M is the absolute permeability of iron, then magnetic flux density of iron ring is​​ 

B =​​ 

B =​​ 

 

 Magnetic flux​​ 

 Magnetic flux​​  = BA​​ 

 Relative permeability of Iron μr

 

     15.   Two  flat  circular  coils  are  made  of  two  identical  wires  each of  length 20cm one  coil  has  number  of  turns  4  and  the  other  2. If the some   current flows though the wire in which will magnetic field at the centre will be greater?

 

Solution​​ 

 For the first coil​​ 

For second coil​​ 

 

 

 

 

 

 

 

 

 

  ​​ Therefore, magnetic field will be greater in coil with 4 turns   ​​ 

     16.  A plat circular coil of 120 turns has a radius of 18cm and carries currents of 3A. What is the magnitude of magnetic field at a point on the axis of the coil at a distance from the centre equal to the radius of the coil?

Solution
Number of turns n = 120

Radius of the coil r = 0.18 m

Axial distance x = 0.18m

Current in coil I = 3A

 

               ​​ B =​​ (4 x 10-7)​​ x 120 x3 x0.182

                             2(0.182 +​​ 0.182)​​ 3/2
​​ 
​​ B= 4.4 x 10-4T

     17.  A current of 5A is flowing upward in a long vertical wire. This wire is placed in a uniform northward magnetic field of 0.02T. How much force and in which direction will this field exert on 0.06 length of the wire?

Solution​​ 

        ​​ 
B = 0.02T

            I = 5A

            L = 0.06

             ​​  = 900
​​ F= 0.02 X 5 X 0.06Sin900​​ 
​​ F = 0.006N

 By Fleming’s Left hand rules the force is directed towards West

     18.   A straight wire of mass 200g and length 1.5m carries a current of 2A. It  is  suspend in  mind  air  by a  uniform  horizontal  magnetic  field  B.  What is the magnitude of the magnetic field?

solution

 M = 200 X 10-3​​ kg​​ 

I = 2A​​ 

l = 1.5m

B =?
F=BIL

Mg = BIL

 B =​​ Mg  ​​ = 200​​ x 10-3​​ x 9.8

       IL                  1.5 X 2

 B = 0.65T

      19.    Two  long  horizontal  wires  are  kept  parallel  at a  distance of  0.2cm  apart  in a vertical plane . both the  wires  have  equal currents  in the  same  direction  the  lower  wire has  a  mass  of  0.05kg/m if the  lower  wire  appears weightless what  is  the  current  in  each  wire ?

 

Solution​​ 

Let  I  amperes be  the  current  in  each  wire the  lower  wire  is  acted upon by  two  forces.

 

 

Since the lower wire appears weightless the two forces were equal over 1m length of the wire

10-4I2​​ = 0.49

     20.   The  horizontal  component  of the  earth magnetic  field  at a  certain  place  is  3 x 10-5​​ and  the  direction  of the  field  is  from  the  geographic  south  to the geographic  North  A very  long straight  conductor  is carrying  a steady current  of  1A. what  is the  force  per unit length on it when it  is  placed  on  a horizontal table  and  the  direction of  the  current  is​​ 

(a)   East  to  West​​ 

(b)   South to  North

 

Solution​​ 

(a)       When  current  is  flowing  from  east  to  west 900

Force on the conductor per unit

 

 

 

(b)      When current  is  flowing from south to  north ​​   =  00​​  ​​ 

    Force on the conductor per unit length​​ 

     21.  A horizontal  straight  wire 5cm long  of  mass 1.2gm-1​​ placed perpendicular  to a  uniform magnetic  field  of  0.6T if  resistance  of the  wire  is  3.85cm-1​​ calculate  the  P.d that  has  to be  applied  between  the  ends  of the  wire  to  make  it just  self supporting ​​ 

 

Solution​​ 

 The  current  (i) in the  wire  is to be in such  a direction  that  magnetic  force  acts on  it  vertically  upward. To  make  the  wire  self  supporting  its  weight  should be equal to the  upward  magnetic  force.

 

Resistance of the wire​​ 

               R = 0.05 x 3.8​​ 

                  = 0.19 

 

Required P. (I)      V = IR​​ 

                             V = 19.6X10-3​​ X 0.19

                             V = 3.7​​ X 10-3V

      22.   A  conductor  of  length  2m carrying  current  of  2A is  held  parallel to  an infinitely long  conductor carrying  current  of  10A  at  a  distance  of  100mm. find  the  force  on  small  conductor​​ 

 

Solution​​ 

 II  = 2A

I2​​ = 10A

r = 100 x 10-3m

l = 2m

Force  on  unit length  of  short  conductor  by the  long  conductor  is  give  by​​ 

   ​​  
Force  on  length  l  =  2m of  short  conductor  by the  long  conductor  is

     ​​ 

   ​​ 

  ​​ 

   ​​ 
The  force  will be  attractive  if  the  direction  of current  is the  same  in  two conduction  and  it will be  repulsive if  the  conductors  carry  current  in the  opposite  directions.​​ 

     23.   In the  figure  below, determine  the  position  between  two wire  which  experience  zero resultant force due to charge   Q  placed  at that  point​​ 

 

 

 

Solution​​ 
The  force  unit  length  acting  in  each wire  of the  parallel  wire  is  given by​​ 

Let​​    be  the  force  per unit  length in  the  wire  carrying  a  current  of  14A ​​ 

         ​​ 

          Since  F1​​ and  F2​​ have  the  same  magnitude  but  they  are  acting in  opposite  direction for  resultant  force  to  be  zero

 

Assume  that  the  charge  Q is  placed  at  a distance  X from the  wire  carrying the​​ 

 

 

 

 

 

 

 

 

 

 

 ​​ 

            The charge   Q  is placed 4m from the either wire.

CLASSIFICATION OF MAGNETIC MATERIALS ​​ 
All substances are  affected  by  magnetic field  some  attain  weak  magnetic  properties and  some  acquire  strong  magnetic  properties  and  some  acquire strong  magnetic  properties.​​ 

              The  magnetic  properties  of the  substances  are  explained  on the  basis  of  modern atomic  theory.​​ 

           The  atoms  that  make  up  any  substance  contain  electrons  that  orbit  around  the  central nucleus.​​ 

           Since  the  electrons  are  charged  they  constitute  an  electric  current and  therefore  produce magnetic  field .

          Thus  an atom behave as  a  magnetic   dipole  and possesses magnetic dipole moment.

           The magnetic  properties  of a  substance  depend upon  the  magnetic  moments  of  its  atoms.​​ 
​​ 

IMPORTANT TERMS USED IN MAGNETISM
The following terms are used in describing the magnetic properties of the materials:

(i)     Magnetic  flux  density  (B)

Is  a  measure  of the  number  of  magnetic  field lines passing  per  unit  area  of  the  material.

 The  greater  the  number  of  magnetic field lines  passing  per  unit  are  of the  material​​ 

            (ii)    Magnetic  permeability​​ 

Is  a  measure  of  its  conductivity  for  magnetic  field  lines​​ 

                                       The  greater the  permeability of  the  material  the  greater  is its   conductivity  for  the  magnetic  field  line  and  vice  versa

 

 Since  magnetic  field  strength B  is  the  magnetic  field  lines passing per  unit  area of the   material, it  is  a measure  of  magnetic  permeability of the  material.​​ 

Suppose magnetic flux density in air or vacuum is. If vacuum/air is replaced by a material, suppose the magnetic flux density in the material becomes B

           Then ratio​​ B/ called the​​ relative permeability   ​​ .   of the material.​​ 

(i)    Relative permeability.
Is the  ratio of  magnetic  flux  density  B in  that  material  to the  magnetic  flux  density ​​  that  would  be if the  material  were replaced by  vacuum/ air .

         Clearly​​ is a  pure  number  and  its  value per  vacuum/air   is  1​​ 

 ​​ Relative  permeability  of  a  material  may also  be  defined  as the  ratio of  absolute permeability  ​​   of  the  material  to  absolute permeability​​ of vacuum/air. 

(ii)   Magnetizing force/ Magnetic intensity ​​ 

Is the number of ampere – turns flowing per unit length   of the toroid.

          The  SI  Unit of  magnetizing force  H is  Ampere – turns  per  meter (AT/m)

 

Consider a toroid with  n turns  per  unit  length carrying  a  current  I. if the  absolute  permeability  of  toroid  material is M, then magnetic  flux  density B in the  material is​​ 

 

The quantity​​  is called magnetizing force or magnetic intensity

Therefore, the ratio​​   in a material I is   from​​ 

; B=

 Thus  if the  some  magnetizing force  is applied to two identical  air  cored  and  iron cored  toroid, then magnetic flux  density  produced inside  the  toroid is​​ 

(iii)    Intensity of magnetization ()   is the magnetic moment developed per unit volume of the material.

           When  a magnetic material  is  subjected  to a  magnetizing  force , the  material  is  magnetized​​ 

          Intensity of magnetization is the  measure  of the  extent to  which  the  material is  a magnetized  and  depends upon the  nature  of the  material

 

   where:​​ 

    ​​ = magnetic moment developed in the material

   V= volume of the material​​ 

If ​​ m​​ is the pole strength developed,​​ 

 is the area of X – section of the material and 2l is the magnetic length. Then​​ 

 

 Hence Intensity  of   magnetization of a  material may  be  defined  as the  pole  strength  developed per  unit area  of  cross – section of the  material.

           Thus  the  SI unit of I is  Am-1  which  is the  same as the  SI  unit of  H​​ 

 Magnetic  susceptibility ​​ is the  ratio  of  intensity  of  magnetic  on I developed in the  material to the  applied  magnetizing  force H​​ 

   ​​ 

       The magnetic susceptibility of a material indicates how easily the material can be magnetized.

          The  unit  of I  is  the  same  as  that of  H so that​​  is  a number​​ 

          ​​ 

 Since  I   is  magnetic  moment per volume​​  is  also  called  volume  susceptibility of  the  material .

Consider a current carrying toroid having core material of relative permeability​​ 

           The  total  magnetic  flux  density  B in the  material  is  given by​​ 

 

Where ​​ 

 = magnetic flux density due to current in the coils.

= magnetic flux density due to the material (Magnetization of the material)
​​         …………………(i)

           …………………(ii)

Here I is the intensity of magnetization induced in the toroid material​​ 

B = +​​ 

Now,​​ 

 
Equation (iii) give the relation between relative permeability (μr​​ ) and magnetic susceptibility (Xm).

CLASSIFICATION OF MAGNETIC MATERIALS

All materials or substances are affected by the external magnetic field. Some attain weak magnetic properties and acquire strong magnetic properties.

On the basis of their behavior in external magnetic field , the various substance classified into the following three categories
(i)Diamagnetic  materials
(ii)Paramagnetic materials​​ 
(iii)Ferromagnetic materials

(i) DIAMAGNETIC MATERIAL
When a diamagnetic substance is placed in a magnetic field in the magnetic field lines prefer to passs through the surrounding air rather than through the substance.


Diamagnetic materials are materials which can not be affected by the magnetic field.

                     They are repelled by magnetic field e.g. lead, silver, copper, zinc, water, gold bismuth etc.

                   These   substances when placed in a magnetic field are weakly magnetized in a direction opposite to that of the applied field.


PROPERTIES OF DIAMAGNETIC MATERIALS

1.   A diamagnetic substance is feebly repelled by a strong magnet.

2.  The magnetic susceptibility () of a diamagnetic substance has a small negative value.

3.   The relative permeability () of a diamagnetic substances is slightly less than 1

4.   When a rod of diamagnetic substances is suspended freely in a uniform magnetic field, the rod comes to rest with its axis perpendicular to the direction of the applied field.

See figure below

 

This gives the relation between relative permeability​​ and magnetic susceptibility​​ of the material.

 

(ii)PARAMAGNETIC MATERIALS​​ 

Are materials which when placed in a magnetic field are weakly magnetized in the direction of the applied field

          The paramagnetic substances include the Aluminum antimony , copper sulphate, Crown grass etc

 Since the weak induced magnetic field is in the direction of the applied field, the resultant magnetic field in the paramagnetic substance is slightly more than the external field

          Hence the magnetic susceptibility of a paramagnetic substance is positive having​​ 

          It clear that the relative permeability​​  for such substances will be slightly more than 1


= 1 +​​ 

  ​​ 

Paramagnetic substance loses its magnetism as soon as the external magnetic field is removed

BEHAVIOR OF PARAMAGNETIC SUBSTANCES IN AN EXTERNAL MAGNETIC FIELD

When a paramagnetic substance is placed in an external magnetic field the dipoles are partially aligned in the direction of the applied field.

 

Therefore the substance is feebly magnetized in the direction of the applied magnetic field. This result into a weak attractive force on the substances.

 

In the absence of the external magnetic field the dipoles of the paramagnetic substances are randomly oriented and therefore the net magnetic moment of the substance is zero.

 

Hence the substance does not exhibit Para – magnetism  ​​ 

PROPERTIES OF PARAMAGNETIC SUBSTANCES​​ 

1.      The relative permeability of a paramagnetic substance is always more than 1

          The result field B inside a paramagnetic substance is more than the external field Bo

2.      The magnetic susceptibility of the paramagnetic substance has small positive value

          It is because​​  and​​ 

3.      The magnetic susceptibility of a paramagnetic material varies inversely as the absolute temperature

 ​​ 

Paramagnetism is quite sensitive to temperature. The lower the temperature the stronger is the paramagnetism and vice versa

4.      A paramagnetic substance is feebly attracted by the strong magnet. It is because a paramagnetic substance develops weak magnetization in the direction of the applied external magnetic field